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CHAPTER 17
Bargaining
Teaching Suggestions
Newspapers often print articles of advice on bargaining for buyers of houses, autos,
and so on. We found a particularly good one that we mention in the text: Andrée Brooks,
“Honing Haggling Skills” (New York Times, December 5, 1993). It mentions (not in technical
language) several points that relate to the concepts in this chapter: best alternative to a negoti -
ated agreement (BATNA), patience, signaling, screening, using mandated agents for credible
commitment, and how multiple dimensions of bargaining facilitate agreement by exploiting
differences in the parties’ relative valuations of the dimensions. You can circulate such an
article to your class a day in advance and start with a discussion that elicits these concepts and
builds them into the game-theoretic framework of the chapter.
You can also look for other popular press publications or Web sites and interpret their
content using the bargaining terminology found in this chapter. Consider the book Getting to
Yes by R. Fisher and W. Ury (Baltimore, Md.: Penguin, 1981) or an Internet site that addresses
bargaining in the process of purchasing a car:
www.edmunds.com/car-buying/negotiating-101.html.
When you cover the alternating-offer models of bargaining, you may find it easier for
your students to follow the progression of the game if they have a simple table showing the
offers and amounts going to each player in each round. If there is gradual decay in the total
value available, we say that the total value drops by x1 after the first offer is rejected and
further rejections lead to drops of x2, x3, . . . (all drops are the same size). For a game where $1
is being split and the value drops by 10 cents after each rejected offer, the last possible offer
would occur when only x10 was left. This bargaining situation can be illustrated with the
following table:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Round Offer by A gets B gets
10 B 0 x10
9 A x9x10
8 B x9x8 + x10
7 A x7 + x9x8 + x10
6 B x7 + x9x6 + x8 + x10
.
.
.
1 A x1 + x3 + x5 + x7 + x9x2 + x4 + x6 + x8 + x10
The final row of the table shows the division that can be anticipated as the Nash equilibrium of
the bargaining game.
For a bargaining game with impatience, let A regard $1 immediately as equivalent to
$(1 + r) > $1 one offer later and B regard $1 immediately as equivalent to $(1 + s) > $1 one
offer later. The values r and s measure impatience. Let a = 1/(1 + r) and b = 1/(1 + s). Then in a
game of dividing $1 that could go, say, 100 rounds, the table of offers and payoffs is as follows:
Round Offer by A gets B gets
100 B 0 1
99 A 1 – b b
98 B a(1 – b) 1 – a + ab
97 A 1 – b + ab – ab2b(1 – a + ab)
96 B a(1 – b + ab – ab2) 1 – a + ab – a2b + a2b2
Whenever A makes an offer, she will ask for
x = 1 – b + ab – ab2 + a2b2 – a2b3 + . . .
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
= (1 – b)(1 + ab + a2b2 + a3b3 + . . .).
The limit, with an indefinite repetition of offers, yields
x = (1 – b)/(1 – ab)
= (s + rs)/(r + s + rs).
Similarly, whenever B makes an offer, she will ask for
y = (r + rs)/(r + s + rs).
Game Playing in Class
You have probably already played some simple bargaining games in the very first
session or two of your class—the take-it-or-leave-it-offer game, the Centipede game, and so
on. Now you can simply remind the class or play some of the games again, compare their
behavior after learning game theory with their behavior before, and discuss this as a matter of
interest in its own right. We set out some options for in-class bargaining over $10 below. You
may also want to try some of the online bargaining games available through Charlie Holt’s
Veconlab site (veconlab.econ.virginia.edu/admin.html).
GAME 1—Bargaining over $10
This is a collection of short bargaining situations that students can be asked to consider
for homework or in class. Variants on these situations are easily created. Instructions: The
minimum offer in all of these situations is 50 cents. Each of you will be paired against the
instructor when determining points. The instructor will respond to your offers using the Nash
equilibrium strategy. If your offer is declined in any of the games, it will be assumed that the
Nash equilibrium outcome is obtained at the next offer.
Situation 1
You and a partner are given $10 to split between you. You make the first move and offer your
partner part of the $10. If she accepts, the bargain is complete and you each get an amount
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
determined by your offer. If she declines, she gets a turn and makes you an offer. Again, the
bargain is complete if you accept, and the split depends on her offer. This process continues,
infinitely if necessary, until one of you makes an offer that the other accepts.
If you are making the first offer in this bargaining game, what do you offer?
_________________
Situation 2
You and a partner are given $10 to split between you. You will make the first move and offer
your partner a portion of the $10. If she accepts, the bargain is complete and you each get an
amount determined by your offer. If she declines, the $10 shrinks to $5 and she then gets a turn
to make you an offer. Again, the bargain is complete if you accept and the division is made
according to the terms of her offer. If you decline her offer, the $5 shrinks to zero, the game
ends, and both of you receive nothing.
If you are making the first offer in this bargaining game, what do you offer?
_________________
Situation 3
You and a partner are given $10 to split between you. You will make the first move and offer
your partner a portion of the $10. If she accepts, the bargain is complete and you each get an
amount determined by your offer. If she declines, the $10 shrinks to $6.67 and she then gets a
turn to make you an offer. Again, the bargain is complete if you accept and the division is made
according to the terms of her offer. If you decline her offer, the sum shrinks to $3.33 and you
get to make an offer. If your partner accepts, the division is made according to your offer, but if
she declines, the sum available shrinks to zero and you each get nothing.
If you are making the first offer in this bargaining game, what do you offer?
_________________
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Situation 4
You are given $10 and told that you will be allowed to make one offer to split the sum with a
partner who will either accept or reject. Regardless of her answer, the game ends after her
move. If she accepts your offer, the $10 is split accordingly. If she rejects it, you both get
nothing.
What amount do you offer your partner in this game? ________________________
Situation 5
You are given $10 and told that you will be allowed to make only one offer to split the sum
with a partner. Your partner must either accept or reject. Regardless of her answer, the game
ends after her move. If she accepts your offer, the $10 is split accordingly. If she rejects your
offer, you get the full $10.
What amount do you offer your partner in this game? ________________________
The results from these bargaining games can be tabulated and given to students for
discussion in class. The first three situations allow discussion of the need to look ahead to later
rounds to consider the best possible first offer. (Variants with impatience can make these more
interesting.) In the last two games, the Nash equilibrium first offer is the same, although many
students do not offer the minimum (here 50 cents) in Situation 4 because of what they identify
as fear of rejection. This provides an additional opportunity to consider the validity of the Nash
equilibrium concept.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
